Electronic calculators have become a common tool for students taking courses at all levels of mathematics. More recently, some of the more sophisticated calculators have also emerged as learning tools. In particular, the features of graphing calculators have resulted in their use in the classroom as they provide significant advantages to the student in the learning process. Graphing calculators, as an example, are characterized by a large screen, which permits the display of mathematical expressions in traditional format, such as with raised exponents and built-up fractions, and also allows multi-lines of information. Here and throughout, the word “expression” is often used to denote equations and inequalities as well as formulas that do not contain an equality or inequality operator.
These graphing calculators also permit displays of graphs, tables and programs. Preferred graphing calculators also permit data transmission to other computing devices, directly or by means of a data storage medium as well as data collection by means of various interface protocols. Many calculator models are designed for particular education levels. However, regardless of the level for which a calculator is designed, a usual goal is to provide a logical and easy to use interface with the student. Two commercially available calculators that are particularly suitable as teaching tools are the Texas Instruments “TI-89” and “TI-92 Plus” Graphing Calculators available from Texas Instruments Incorporated of Dallas, Tex.
Mathematics is particularly suited to obtaining solutions to problems by correctly selecting a proper transformation or operation rule and then applying the selected rule to the problem. More specifically, this process represents a programming paradigm often classified as “rewrite rules.” Further, in a more general sense, rewrite rules may also be applied to other uses such as optimizing compilers, parsing natural and computer languages, database queries, theorem proving, and especially computer CAS (Computer Algebra Systems). This paradigm is also called term rewriting, or rule-based programming and is related to equational logic and constraint-based programming. Most of the literature on this subject may be found in the references titled, “Proceedings nth Rewriting Techniques and Applications,” for n=1, 2, etc., published by Springer-Verlag.
Although as discussed above, even though “rewrite rule” paradigms have other applications, they are particularly applicable to mathematics. Consequently, the discussions herein are made with respect to mathematics and more specifically with respect to areas of mathematics for which CAS (Computer Algebra Systems) have been developed and used with computers and calculators.
An example of a “rewrite rule” or transformation that may be applied to trigonometry is:For all A, sin(A)/cos(A)→tan(A)
As an example of its use, this rule could transform5+sin(3x)/cos(3x)+sin(y2)/cos(y2)to5+tan(3x)+tan(y2).As is clear, in this example A matches 3x in the second term, whereas A matches y2 in the third term.
To help avoid confusion with respect to the above example as well as other examples used herein, capital letters will be used to represent “pattern variables” (e.g. A), and lower case letters will be used to represent the user variables (e.g. x and y).
Furthermore, some rewrite rules might have certain conditions that the match must satisfy for the replacement to occur. For example, for a rule related to differentiation, the rule might be:For all A, U and variables X such that A is free of X, d/dX(A·U)→A d/dX(U).
Furthermore, there are often several alternatives for applying an applicable rule. For example, one extreme is to apply a rule only once and only if the pattern matches the entire expression. The other extreme is to apply the rule repeatedly wherever it is applicable throughout the expression until the expression is idem potent, which means the rule is no longer applicable anywhere in the expression.
Most CAS (Computer Algebra Systems) are intended for mathematically experienced users. Therefore, such systems typically manipulate large mathematical expressions and produce simplified final answers without showing any of the intermediate steps. These types of computer algebra systems may provide rewrite rules as a convenient way for users to extend the built-in mathematical capabilities of their computer or calculator. For example, a user can use these capabilities to implement Bessel functions together with their derivatives, integrals and recurrence relations if these mathematical capabilities are not already built-in or available as an add-on package. However, the built-in mathematical capabilities are typically implemented entirely, or at least substantially by using functional and/or procedural paradigms, which execute fast and are more suitable for highly synthetic algorithms such as modern polynomial factoring algorithms. These type algorithms do nothing to aid a student in learning the process and are typically most appropriate for mathematically experienced users.
In contrast, and according to the present invention, a CAS is intended to help a student learn subjects such as algebra, pre-calculus, calculus, or any other area of mathematics and is based on a different set of targets or goals. For example, a CAS used for teaching should allow the user to highlight the expression that is to be transformed so that students can easily revise an earlier step. Furthermore, some mathematical expressions are extremely complex and are simplified by transforming only certain sub-expressions. Therefore the system should also allow the user to highlight sub-expressions that are only part of an entire expression. Thus, at each step, the student may control exactly where transformations or operations are carried out. It should be noted that both the terms “transformation” and “operation” are used herein and are substantially synonymous. However, transformation is somewhat more appropriate for the context of rewrite rules, and consequently may be used herein to describe any operation including non-mathematical operations. The desired system could also automatically choose the sub-expression and transformation for the next step or alternately do this for all of the successive steps in a derivation without user intervention. These goals encourage the use of rewrite rules to implement almost all the transformations for such educational computer algebra systems. This is because rewrite rules: (a) most clearly reflect the way students are taught to do derivations; (b) such rules are inherently modular and facilitate the selection of various subsets of transformations that can be applied to each step of the derivation; and (c) separating the transformations into left side patterns and right side replacements facilitates building a context-sensitive menu of applicable rules that can then be presented to the user.
As will be appreciated by those skilled in the art, there are several hundred rules that are applicable to the teaching of algebra through calculus. Unfortunately, because of the wide range of skills and applicability, use of the rules often entails several difficulties. For example, dozens of transformations are often applicable to an expression. Unfortunately, menus offering more than just a few choices may be extremely intimidating, difficult, and even worthless to students who are just learning the techniques that are addressed or selectable from the various menu items. In addition, many of the rules will reverse the operation of previous applied rules, and therefore, can lead to an infinite loop if both of the rules are repeatedly and alternately applied. For example, consider the rules       A          -      N        →                    1                  A          N                    ⁢                          ⁢              and            ⁢                          ⁢              B                  A          N                      →          B      ·                        A                      -            N                          .            Both of these transformations are applicable to the expression       x          -      2        +            3              x        5              .  However, as one skilled in the art will quickly understand, if both of these rules are applied until neither is applicable, they keep reversing each other until the replacements are aborted either by exhaustion of memory or by a keyboard interrupt from the user.
Still another difficulty arises from the fact that it is sometimes desirable to employ one set of rules to trigger a particular transformation, but employ a different set of rules to further transform a new sub-expression generated by the triggering transformations. For example, considering the rule AN·AM→AN+M for collecting like factors. Acting alone, this rule would transform x2·x3+6+1 to x2+3+6+1, which would be very helpful to weak students or students who are just learning how to use this collection rule. However, for more advanced students it is desirable to have a transformation “combine like factors” that would instead transform the expression to x5+6+1 by also doing the arithmetic in the exponent generated by collecting like factors. However, even when “combine like factors” is used, gratuitous arithmetic is not also automatically outside the newly generated exponent. For example, the 6+1 is not automatically added together to yield 7 in the above example.
Thus, it should be recognized that as students start learning a particular subject or set of rules with respect to mathematics, it is appropriate that very small or fine steps be used as they learn to do typical algebraic simplifications. However, coarser or more involved automatic algebraic simplifications are appropriate to the student who already knows the rules of algebraic simplification quite well and is trying to learn how to solve a linear equation. Similarly, as a student's ability increases, coarser steps for solving a linear equation would better serve the student when learning how to solve a quadratic equation. Likewise, coarser algebraic simplification steps for solving quadratic equations should be used when the student is learning differentiation and/or symbolic integration. Therefore, it should be clear that the most appropriate menu items do not depend only upon the users expression that is to be transformed, but also the problem area and the skill of the student learning from the process.
Another difficulty is that the usual practice of applying rewrite rules may employ a very inefficient algorithm that can be intolerably slow for large problems.
Prior Art
MathPert™ is a program available from Mathpert Systems, 2211 Lawson Lane, Santa Clara Calif. 95054. The principal program author M. Beeson describes the implementation in an article titled “Design Principles of MathPert™”, published in the conference proceedings “Computer-Human Interaction and Symbolic Computation”, (editor N. Kajler, Springer-Verlag, ISBN 3-211-82843-5, pages 89–115). The article includes a brief reference to rewrite rules, but the article strongly suggests that they are not being used in the manner of the present invention.
The HP-40G, HP-48G, and Casio FX 2.0 hand-held calculators have some very limited computer algebra that allows students to choose or show only large steps such as “factor”, “expand”, or “common denominator”. Consequently, there are so few different possible transformations that there is little need to build a context-dependent menu or provide for subsets of more than one transformation.